ON UPPER ESTIMATES OF THE PARTIAL-SUMS OF A TRIGONOMETRIC SERIES IN TERMS OF LOWER ESTIMATES

Let {a(k)}k=0infinity and {b(k)}k=0infinity be sequences of real numbe rs and let S(n)(x) be defined by S(n)(x) = SIGMA(k=0)n(a(k) cos(kx) b(k) sin(kx)), n = 0, 1,... . It is proved that the estimate max(x) S( n)(x) less-than-or-equal-to 4a0n1-alpha holds for each natural number n such that S(m)(x) greater-than-or-equal-to 0 for all x and m = 1,... , n. Here alpha is-an-element-of (0, 1) is the unique root of the equa tion integral0(3pi/2) t(-alpha) cos t dt = 0. It is proved that the or der n1-alpha in this estimate cannot be improved. Various generalizati ons of this result are also obtained. Bibliography: 10 titles.